Problem: The people of Bridgetown wanted to build a bridge across a nearby river. Since they were poor swimmers, their master Trigonomos agreed to measure the width of the river without actually crossing it. Trigonomos spotted a tree across the river and marked the spot directly across from it. Then he walked to another point $10$ meters down the river and found that the angle between his side of the river and the line connecting him to the tree was $40^\circ$. What is the width of the river? Round your final answer to the nearest hundredth.
The strategy Model the situation as a right triangle. Determine the appropriate trigonometric ratio in order to find the missing side. Form an equation and solve for the missing side. Calculate the final result and round. Modeling as a right triangle This situation can be modeled by the following right triangle. The base is $10\text{ m}$ and the angle on the right is $40^\circ$. We are asked to find the width of the river, which is the height of the triangle. ${40^{\circ}}$ $?$ $10$ Determining the appropriate trigonometric ratio We are given the measure of an angle and the length of the side ${\text{adjacent}}$ to the given angle. We are asked to find the side ${\text{opposite}}$ to the given angle. The appropriate trigonometric ratio is therefore the $\text{tangent}$. Forming an equation and solving Denoting the missing side by $x$, we obtain the equation $\tan(40^\circ)=\dfrac{x}{10}$. Solving the equation, we get $x=10\cdot\tan(40^\circ)$. Evaluating this result in the calculator and rounding to the nearest hundredth, we get $x=8.39\text{ m}$. Summary The width of the river is $8.39$ meters.